Cardiac myofibers are densely packed in the heart wall and are locally aligned to helical curves. Helices act as geodesics between points in the myocardium and mathematical analyses support the view that this alignment is mechanically optimal. As a result, geometric descriptions of cardiac fibers using the helix angle, taken to be the projected angle between the fiber direction and the short-axis plane (see first image 100A in FIG. 1), are popular within the prior art. Several accounts from both small-scale histology and voxel-scale studies based on Diffusion MRI (dMRI) report that along a transmural penetration line from the heart's outer to inner wall, the helix angle varies smoothly and regularly undergoing a total change in orientation of about 120°. The range of the transverse angle, which is the angle formed by a fiber moving away from a plane perpendicular to the transmural direction, is much smaller, about ±10°, and is therefore often ignored in the literature,
The analysis of myofibers from histological slices is cumbersome and their invasiveness does not easily admit an association with the original intact three-dimensional geometry. Thus, many modern analysis methods work with cardiac fiber orientation data derived from dMRI measurements. However, the scale at which current dMRI measurements are made is at least one order of magnitude larger than the length of individual cardiomyocytes The measured signal therefore reflects the composite behaviour of large groups of cardiac muscle cells within the collagen matrix (see third image 100C in FIG. 1). Within the prior art a promising characterization of the collective geometrical variation of cardiac fibers, using a method derived from texture flow analysis, has been reported. This work concluded that that the cardiac fiber directions across three mammalian species, the rat, the canine and the human, are locally described by a particular minimal surface, the generalized helicoid model (GHM).
However, a limitation of the GHM is that its streamlines lie on a planar manifold in spite that the heart wall is curved (see first image 100A in FIG. 1). The GHM thus captures the variability of cardiac fibers in a plane tangent to the local cardiac wall but not orthogonally to it. Moreover, experimental results have shown that the GHM is only accurate in the immediate neighborhood of a voxel, with fitting errors growing rapidly as the neighborhood in which the fits are applied is increased.
Accordingly it would be beneficial to provide an alternate model for modeling cardiac fibers within heart walls that removes the limitations of GHM.